Delete specific fcurve in python
As you know the scenes animation fcurves are found in

fc = bpy.context.scene.animation_data.action.fcurves


The fcurves list has a remove method -

fc.remove(fc[0])


Which leads to changing your code to

for scene in bpy.data.scenes:
    fcurves = scene.animation_data.action.fcurves
    for c in fcurves:
        if c.data_path.startswith("animated_render_border"):
            fcurves.remove(c)